Calabi flow in Riemann surfaces revisited: a new point of view

In this paper, we observe a set of functionals of metrics which all decrease under the Calabi flow and have uniform lower bound along the flow, which gives rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we prove the long-term existence and convergence of the Calabi flow. Thus we give a new proof to Chruscial's theorem. The set of simple ideas of global integral estimates and concentration compactness should have further implications in other heat flow problems.